Modules whose surjective endomorphisms have a $\gamma$-small kernels

Document Type : Research Paper


1 Department of Mathematics, Faculty of Sciences, University of Mohammed First, Oujda, Morocco

2 Department of Mathematics, Faculty of Sciences, University of Mohammed First, Oujda, Morocco.


In this paper, we introduce a proper generalization of that of Hopfian modules, called $\gamma$-Hopfian modules. A right $R$-module $M$ is said to be $\gamma$-Hopfian, if any surjective endomorphism of $M$ has a $\gamma$-small kernel. Some basic characterizations of $\gamma$-Hopfian modules are proved. We prove that a ring $R$ is semisimple cosingular if and only if every $R$-module is $\gamma$-Hopfian. Further, we prove that the $\gamma$-Hopfian property is preserved under Morita equivalences. Some other properties of $\gamma$-Hopfian modules are also obtained with examples.


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